The Connection Principle (hereafter, CP) says that there is some kind of internal relation between a state's1 having intentional content ("aspectual shape") and its being (at least potentially) conscious. Searle's argument for the principle is just that potential consciousness is the only thing he can think of that would distinguish original intentionality from ersatz (Searle, 1992, pp. 84, 155 and passim. All Searle references are to 1992). Cognitivists have generally found this argument underwhelming given the empirical successes recently enjoyed by linguistic and psychological theories with which, according to Searle, CP is not reconcilable. Our primary interest in this paper is not, however, to decide whether CP is true, but just to get as clear as we can about what exactly it asserts. Finding a reasonable formulation of the principle turns out to be harder than Searle appears to suppose; or so we claim.